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De Morgan's Laws for the Intersections and Unions of Sets

Recall from The Relative Complement and Complement of a Set page that if $A$ and $B$ are sets then the relative complement of $A$ with respect to $B$ is $B \setminus A = \{ x : x \in B \: \mathrm{and} \: x \not \in A \}$ and the complement of $A$ with some universal set $U$ in mind is $A^c = \{ x : x \not \in A \}$.

We will now look at some very important theorems regarding the complements of sets that were discovered by Augustus De Morgan and are famously known as De Morgan's laws.